![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupequzmptf | Structured version Visualization version GIF version |
Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
limsupequzmptf.j | ⊢ Ⅎ𝑗𝜑 |
limsupequzmptf.o | ⊢ Ⅎ𝑗𝐴 |
limsupequzmptf.p | ⊢ Ⅎ𝑗𝐵 |
limsupequzmptf.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
limsupequzmptf.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
limsupequzmptf.a | ⊢ 𝐴 = (ℤ≥‘𝑀) |
limsupequzmptf.b | ⊢ 𝐵 = (ℤ≥‘𝑁) |
limsupequzmptf.c | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) |
limsupequzmptf.d | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
limsupequzmptf | ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1915 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | limsupequzmptf.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | limsupequzmptf.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
4 | limsupequzmptf.a | . . 3 ⊢ 𝐴 = (ℤ≥‘𝑀) | |
5 | limsupequzmptf.b | . . 3 ⊢ 𝐵 = (ℤ≥‘𝑁) | |
6 | limsupequzmptf.j | . . . . . 6 ⊢ Ⅎ𝑗𝜑 | |
7 | limsupequzmptf.o | . . . . . . 7 ⊢ Ⅎ𝑗𝐴 | |
8 | 7 | nfcri 2943 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴 |
9 | 6, 8 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐴) |
10 | nfcsb1v 3852 | . . . . . 6 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐶 | |
11 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑗𝑉 | |
12 | 10, 11 | nfel 2969 | . . . . 5 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉 |
13 | 9, 12 | nfim 1897 | . . . 4 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉) |
14 | eleq1w 2872 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴)) | |
15 | 14 | anbi2d 631 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐴) ↔ (𝜑 ∧ 𝑘 ∈ 𝐴))) |
16 | csbeq1a 3842 | . . . . . 6 ⊢ (𝑗 = 𝑘 → 𝐶 = ⦋𝑘 / 𝑗⦌𝐶) | |
17 | 16 | eleq1d 2874 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐶 ∈ 𝑉 ↔ ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉)) |
18 | 15, 17 | imbi12d 348 | . . . 4 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉))) |
19 | limsupequzmptf.c | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐶 ∈ 𝑉) | |
20 | 13, 18, 19 | chvarfv 2240 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑉) |
21 | limsupequzmptf.p | . . . . . . 7 ⊢ Ⅎ𝑗𝐵 | |
22 | 21 | nfcri 2943 | . . . . . 6 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐵 |
23 | 6, 22 | nfan 1900 | . . . . 5 ⊢ Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ 𝐵) |
24 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑗𝑊 | |
25 | 10, 24 | nfel 2969 | . . . . 5 ⊢ Ⅎ𝑗⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊 |
26 | 23, 25 | nfim 1897 | . . . 4 ⊢ Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ 𝐵) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊) |
27 | eleq1w 2872 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐵 ↔ 𝑘 ∈ 𝐵)) | |
28 | 27 | anbi2d 631 | . . . . 5 ⊢ (𝑗 = 𝑘 → ((𝜑 ∧ 𝑗 ∈ 𝐵) ↔ (𝜑 ∧ 𝑘 ∈ 𝐵))) |
29 | 16 | eleq1d 2874 | . . . . 5 ⊢ (𝑗 = 𝑘 → (𝐶 ∈ 𝑊 ↔ ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊)) |
30 | 28, 29 | imbi12d 348 | . . . 4 ⊢ (𝑗 = 𝑘 → (((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) ↔ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊))) |
31 | limsupequzmptf.d | . . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐵) → 𝐶 ∈ 𝑊) | |
32 | 26, 30, 31 | chvarfv 2240 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → ⦋𝑘 / 𝑗⦌𝐶 ∈ 𝑊) |
33 | 1, 2, 3, 4, 5, 20, 32 | limsupequzmpt 42371 | . 2 ⊢ (𝜑 → (lim sup‘(𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶)) = (lim sup‘(𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶))) |
34 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
35 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑘𝐶 | |
36 | 7, 34, 35, 10, 16 | cbvmptf 5129 | . . . 4 ⊢ (𝑗 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶) |
37 | 36 | fveq2i 6648 | . . 3 ⊢ (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶)) |
38 | 37 | a1i 11 | . 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐴 ↦ ⦋𝑘 / 𝑗⦌𝐶))) |
39 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑘𝐵 | |
40 | 21, 39, 35, 10, 16 | cbvmptf 5129 | . . . 4 ⊢ (𝑗 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶) |
41 | 40 | fveq2i 6648 | . . 3 ⊢ (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶)) |
42 | 41 | a1i 11 | . 2 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶)) = (lim sup‘(𝑘 ∈ 𝐵 ↦ ⦋𝑘 / 𝑗⦌𝐶))) |
43 | 33, 38, 42 | 3eqtr4d 2843 | 1 ⊢ (𝜑 → (lim sup‘(𝑗 ∈ 𝐴 ↦ 𝐶)) = (lim sup‘(𝑗 ∈ 𝐵 ↦ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 ⦋csb 3828 ↦ cmpt 5110 ‘cfv 6324 ℤcz 11969 ℤ≥cuz 12231 lim supclsp 14819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ico 12732 df-limsup 14820 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |